The Physical Aspects of Fish Locomotion
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THE PHYSICAL ASPECTS OF FISH LOCOMOTION BY E. G. RICHARDSON, B.A., PH.D., D.SC. (Armstrong College, Newcastle on Tyne.) (Received January i, 1935.) (With Eight Text-figures.) I. INTRODUCTION. THE mechanism by which a fish is able to dart through a heavy medium like water with little apparent effort has long been a matter of interest to zoologists. Before the arrival of the cinematograph in its modern form, Marey (1884) obtained a series of " chronophotographic" records of fishes in movement to demonstrate the sinusoidal motion, while more recently Magnan (1930) and Gray (1933) have taken cinematograph pictures at the rate of 10-20 per sec, in which the fish is seen swim- ming behind a reticule, thus making detailed study of the mode and rate of pro- gression possible. Houssay (1912) and Breder (1926) have also made contributions to this problem. Their work will be noted later. With the general adoption of streamlined form for all bodies which have to move rapidly through air or water, the attention of workers in hydrodynamics has been focused on the same problem. Indeed, the extensive researches of Magnan were subventioned by the French Air Ministry. It is from this aspect that the author is interested in the subject. There appear to be two possible explanations to account for the rapidity of movement of a fish through the water, viz. (1) that in virtue of the mucus on its surface it may possess a resistance lower than that of the same shape when unlubricated, (2) that in virtue of its flexibility it cleaves the water with less re- sistance than if rigid. If either of these facts can be substantiated it is obviously of great importance to aircraft and ship design. II. RESISTANCE MEASUREMENTS. Although Houssay made measurements of the resistance of wooden models of fish towed through the lake of the ficole Militaire at Paris, the only measurements on fishes were those of Magnan deduced from cinema photographs of the weighted bodies of fish freely falling in a water tank. The photographs are scarcely clear enough for the exact deduction of the rate of fall. Indeed Magnan decides that the space : time curves are exact parabolas, which as he rightly deduces, implies that the resistance is independent of the velocity. He does not, however, complete the argument which is that if the parabolic law is followed the velocity increases without limit, a condition only fulfilled on free fall in a vacuum. A body falling in a 64 E. G. RICHARDSON resisting medium must sooner or later reach a constant velocity, determined by the balance between gravity and resistance, which can then be determined as a function of the terminal velocity. The author decided to determine the rate of fall of fishes over various distances by a direct chronographic method. A tank 8 ft. high and 1 ft. 6 in. square was filled with water and equipped with starting and stopping devices for the object whose fall was to be investigated. The former device was arranged so that the fish could be started from a little below the surface, the latter was a tray which could be lowered by cords passing over pulleys to counterpoise weights. The fish was held in an electromagnetic grip by its tail. Breaking the electric circuit released the fish and simultaneously released the brake of a Wood's chronograph. When the nose of the fish hit the lower stage, which was a pivoted gauze network, it broke a contact, which stopped the chronograph again. As will be seen from the diagram of the apparatus (Fig. 1) this contact had to be arranged in a vessel above water, otherwise it did not break smartly, the water being a conductor of electricity. With this apparatus the time could be read to 0-002 sec, while the distances were measured by telescope to 1 mm. The freshly killed fish, deprived of all its fins except the caudal fin, was loaded by lead shot in the mouth. Trouble was experienced at distances of fall exceeding 2 ft. owing to the flexibility of the fish which often led it to describe a curved trajectory. Later a wire stiffener was pushed into the body to keep the fall vertical. At any given distance, times were repeated until a minimum had been established. After each fall the lower stage was raised to retrieve the fish. When a series of falls at various distances and weights had been made, the fish was carefully measured and a template made. Subsequently a wooden model was made to the template, loaded in the scooped-out interior by lead shot and provided with an aluminium tail. The time of fall of the model in the tank was then determined under three conditions of surface, i.e. (a) roughened, (b) smoothed and polished, (c) greased with oil. Writing the equation of motion of the falling body in the form , m dt=mg~R' where the first term represents the downward force on the body at the instant when its velocity is v, the second term is the weight of the body in the water, or buoyancy, and the third is the resistance. We will write the latter as a function of v in the form kvn, so that , m =m g + kvn. at n may have, in practice, values between o and 2; (a) if n = o, v = ( ) t, the \m ml acceleration is constant and we have the case postulated by Magnan; (b) if n = 1 (Stokes' law), fc the terminal velocity is m'glmk; (c) if n = 2 (Newton's law), v = c tanhgjc, where c is the terminal velocity =Vm'glmk. In case (c) the terminal velocity is of course less, The Physical Aspects of Fish Locomotion 65 ceteris paribus, and reached in a shorter time. On Fig. 2 are plotted the displace- ment : time curves for a mackerel and a herring loaded by different weights to show that a terminal velocity, indicated by the graph becoming a straight line, is in To starting switch of chronograph [*o stopping switch of chronograph Fig. 1. Tank for time of fall of fishes and models. practically every case attained within the depth of the tank. Probably Magnan's failure to detect a terminal velocity is due to his not having carried the distances far enough. (Although he had at his disposal a tank 3 metres deep, the displacement: time curves on which he bases his n = o deduction seem not to extend beyond half a metre.) Table I gives the resulting data in the form of weight in water (W) and corresponding terminal velocity (V). These are subsequently plotted in the form JEB-XIIli 5 66 E. G. RICHARDSON log W: log V on Fig. 3. The slope of these curves gives the appropriate value of n at each value of V. 3 4 5 Time (sees.) Fig. 2. Rate of fall of fishes Table I. Resistance measurements. Mackerel. Weight in air 1OC3 gm., in water 267 gm., volume 123 c.c. W gm. 267 40-5 480 650 Fcm./sec 6-65 n-o 125 13-5 Rough wooden model. Weight in air 151 gm., in water 267 gm., volume 130 c.c. W 267 41-5 48-2 65-0 V 83 100 i2-o 13-3 Wooden model smoothed and varnished. W 267 41 -5 45-0 V 8-3 12-0 12-5 Herring. Weight in air 864 gm., in water 13-8 gm., volume 72-6 c.c. W 13-8 230 31-2 V S'O 100 12-0 Rough wooden model. Weight in air 84-2 gm., in water 13-9 gm., volume 70-3 c.c. W 13-9 23-0 31-0 V 5-0 9-0 no Wooden model smoothed and varnished. W 13-9 188 23-0 31-3 V 5"25 8'0 IO-O I2'2 The general conclusion to be drawn from this table and the graphs is that the fish has a slightly less resistance than the rough wooden model, but that a carefully smoothed and polished wooden model can improve on this, particularly at the low speeds. The exponent n given by the slope of the curves of Fig. 3 varies from i-o to i-8 for the fishes used, just below the turbulence regions to which Newton's law applies. It is probable that the normal velocity range covered by a fish lies below the turbulent regime except when it is spurred to violent movement. The Physical Aspects of Fish Locomotion 67 The experiments with the smoothed model covered with a light oil previous to immersion to imitate the mucus on the body of the fish did not indicate any marked • fish o rough model X varnished model 1-0 1-2 1-0, 0-6 0-8 1-0 1-2 1-4 log terminal V Fig. 3. Resistance of fishes. +4 Paraffin +2 I Interface between / / / liquid -2 0 I 2 Velocity (cm./sec.) Fig. 4. Velocity gradients in two liquids in relative motion. improvement on the resistance of the model at speeds below the critical. Both theory and practice, however, concur in indicating that the presence of a viscous layer, i.e. more viscous than the main body of the fluid in which it is immersed, 5-2 68 E. G. RICHARDSON delays the onset of turbulence. It is in fact the shearing force in the "boundary layer " of fluid close to the solid surface of an immersed obstacle that gives rise to instability which in turn leads to turbulence in the wake.